Question

R be a relation on Z defined by R= {(a,b): a-b is an integer} show that R is an equivalence relation​

Answer 1

$\textbf{Given:}$

$aRb\;{\iff}\;\text{a-b is an integer}$

$\textbf{To prove:}$

$\text{R is an equivalence relation}$

$\textbf{Solution:}$

$\text{Let a{\in}Z }$

$\implies\,a-a=0\;\text{which is an integer}$

$\implies\,aRa$

$\therefore\,\textbf{R is reflexive}$

$\text{Let aRb }$

$\implies\,a-b\;\text{is an integer}$

$\implies\,b-a\;\text{is also an integer}$

$\implies\,bRa$

$\therefore\,\textbf{R is symmetric}$

$\text{Let aRb and bRc }$

$\text{Then,}\;a-b=k\;\;\text{and}\;\;b-c=l\;\;\text{where k,l are integers}$

$\text{Adding we get,}$

$(a-b)+(b-c)=k+l$

$\implies\,a-c=k+l\;\text{which is an integer}$

$\implies\,aRc$

$\therefore\textbf{R is transitive}$

$\textbf{Hence R is an equivalence relation}$